To protect plants from heat, a shed of iron rods covered with green cloth is made. The lower part of the shed is a cuboid mounted by semi-cylinder as shown in the figure. Find the area of the cloth required to make this shed, if dimensions of the cuboid are \(14 \text{ m} \times 25 \text{ m} \times 16 \text{ m}\).
Show Hint
Always visualize the surfaces covered. In shed problems, the floor is usually excluded. Also, ensure the radius and length of the semi-cylinder correctly correspond to the cuboid's dimensions from the diagram.
Step 1: Understanding the Concept:
The shed consists of a cuboidal base and a semi-cylindrical top.
The cloth covers the four vertical walls of the cuboid and the curved surface of the semi-cylinder, including its two semi-circular ends. The bottom of the shed (floor) is not covered with cloth. Step 2: Key Formula or Approach:
Area of cloth = Lateral Surface Area of Cuboid + CSA of Semi-cylinder + Area of 2 Semi-circles.
Dimensions: Length (\(L\)) = \(25 \text{ m}\), Width (\(W\)) = \(14 \text{ m}\), Height (\(H\)) = \(16 \text{ m}\).
For the semi-cylinder: Radius (\(r\)) = \(\frac{W}{2} = 7 \text{ m}\), Length (\(h_{cyl}\)) = \(25 \text{ m}\). Step 3: Detailed Explanation:
1. Lateral Surface Area of cuboid (4 walls):
\[ \text{LSA} = 2(L + W) \times H = 2(25 + 14) \times 16 = 2 \times 39 \times 16 = 1248 \text{ m}^2 \]
2. Curved Surface Area of semi-cylinder:
\[ \text{CSA} = \frac{1}{2}(2 \pi r h_{cyl}) = \pi r h_{cyl} = \frac{22}{7} \times 7 \times 25 = 550 \text{ m}^2 \]
3. Area of 2 semi-circular ends:
\[ \text{Area}_{\text{ends}} = 2 \times \frac{1}{2} \pi r^2 = \pi r^2 = \frac{22}{7} \times 7^2 = 154 \text{ m}^2 \]
Total Area = \(1248 + 550 + 154 = 1952 \text{ m}^2\). Step 4: Final Answer:
The total area of the cloth required is \(1952 \text{ m}^2\).
